Eigenstates for a Hydrogen Atom Confined to an Infinite Spherical Potential Well

Apart from its intrinsic interest, the problem of a confined hydrogen atom was motivated by an attempt to model the effects of high pressure on a physical system. It is also relevant to quantum dots and white dwarfs. Among the many approaches to the problem have been numerical solutions, perturbation calculations and variational methods [1–3].

The exact solution for a free hydrogen atom can be represented by a wavefunction of the form

,

where is the reduced radial function, which can be expressed in terms of a Whittaker function:

.

This is a solution of the radial Schrödinger equation (in atomic units):

.

In this Demonstration, we consider an approximation for the hydrogen atom confined to an infinite spherical potential well of radius in the form

,

where is a parameter that you can vary. The radial functions obey the boundary condition . We consider the states with principal quantum numbers to .

The left-hand plots in the graphic show the radial functions (red curves) for selected values of and . Values of up to 30 hartrees are considered. The free hydrogen atom radial functions (black curves) are also plotted for reference.

The right-hand plots show the energies , in hartrees, as a function of . As becomes large, the free hydrogen atom energies are approached. For sufficiently small values of , the energies become positive, as the effect of confinement overcomes the Coulomb attractive energy.